A statistical-mechanical formalism of chaos based on the geometry of invariant sets in phase space is discussed to show that chaotic dynamical systems can be treated by a formalism analogous to that of thermodynamic systems if one takes a relevant coarse-grained quantity, but their statistical laws are quite different from those of thermodynamic systems. This is a generalization of statistical mechanics for dealing with dissipative and hamiltonian (i.e., conservative) dynamical systems of a few degrees of freedom.Thus the sum of the local expansion rate of nearby orbits along a relevant orbit over a long but finite time has been introduced in order to describe and characterize (1) a drastic change of the structure of a chaotic attractor at a bifurcation and anomalous phenomena associated, (2) a critical scaling of chaos in the neighborhood of a critical point for the bifurcation to a nonchaotic state, and a self-similar temporal structure of a critical orbit on the critical 2~ attractor and the critical golden tori without mixing, (3) the critical KAM torus, diffusion and repeated sticking of a chaotic orbit to a critical torus in hamiltonian systems. Here a q-phase transition, analogous to the ferromagnetic phase transition, plays an important role. They are illustrated numerically and theoretically by treating the driven damped pendulum, the driven Duffing equation, the Henon map, and the dissipative and conservative standard maps.This description of chaos breaks the time-reversal symmetry of hamiltonian dynamical laws analogously to statistical mechanic!~ of irreversible processes. The broken timereversal symmetry is brought about by orbital instability of chaos.

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Intermittent chaos exhibits regular laini~ar motions and irregular turbulent bursts alternately, indicating that its chaotic attractor has two different types of local structures. For type I intermittency just before the saddle-node bifurcation, iUs shown that the two types of local structures can be captured by the fluctuation spectrum h(A) of the coarse-grained local expansion rates A of nearby orbits and their q-weighted average A(q), (-=O and A=Az=Owith a slope q., leading to a discontinuous transition of A(q) from Al to Az at q=q. as q is increased across q.. This represents a phase transition between the laminar motions and the turbulent bursts, and gives a new type of q-phase transition with 1.0 >q.>0.5 in contrast to other three types of q-phase transitions with transition points qa=2.0, qp<0.5 and qr=1.0. Thus statistical mechanics of chaotic attractors at the bifurcation points is fully developed for the nontrivial case. § 1. Introduction Many efforts have been made in order to construct statistical mechanics, of chaotic dynamical systems which is useful for exploring non-equilibrium open systems far from equilibrium_1)-3) Very recently, important physical quantities have been . ., discovered; for example, the singularity spectrum I(a) of the natural invariant measure,4) and the fluctuation spectrum h(A) of the coarse-grained local expansion rates A of nearby orbits along the local unstable manifolds. 5 )-7)' They are the coarse-grained quantities which correspond to the thermodynamic functions in statistical thermodynamics. These quantities are the scaling exponents which describe the scaling structures of chaotic attractors, and lead to the variational principles for the thermodynamic formalism of chaotic dynamical systems which relate various scaling structure functions to each other within each class. 2 ),4)-7) The spectra I(a) and h(A), however, belong to different classes so that their relation is beyond the thermodynamic formalism. 7 ).The next step is, therefore, to find a statistical-mechanical formalism of the scaling structures in terms of the orbital structures of chaotic attractors. Then it would be important to study a relevant bifurcation of chaos which brings about an eminent change of a chaotic attractor at the bifurcation point. The purpose of the present series of papers 8 ),9) is, therefore, to characterize the chaotic attractors at the bifurcation points for various bifurcations by the scaling structures, and then elucidate the scaling structures from the orbital structures of the critical attractors.In the present paper, we shall study a chaotic attractor just before the saddle-node bifurcation from the above viewpoint. Such a chaotic attractor is known to exhibit

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